Fock representation of the renormalized higher powers of white noise and the Virasoro--Zamolodchikov--w∞∗w_{\infty} *--Lie algebra

The identification of the $*$--Lie algebra of the renormalized higher powers of white noise (RHPWN) and the analytic continuation of the second quantized Virasoro--Zamolodchikov--$w_{\infty} *$--Lie algebra of conformal field theory and high-energy physics, was recently established in \cite{id} based on results obtained in [1] and [2]. In the present paper we show how the RHPWN Fock kernels must be truncated in order to be positive definite and we obtain a Fock representation of the two algebras. We show that the truncated renormalized higher powers of white noise (TRHPWN) Fock spaces of order $\geq 2$ host the continuous binomial and beta processes

Quantum Markov fields on graphs

We introduce generalized quantum Markov states and generalized d-Markov chains which extend the notion quantum Markov chains on spin systems to that on $C^*$-algebras defined by general graphs. As examples of generalized d-Markov chains, we construct the entangled Markov fields on tree graphs. The concrete examples of generalized d-Markov chains on Cayley trees are also investigated.Comment: 23 pages, 1 figure. accepted to "Infinite Dimensional Anal. Quantum Probability & Related Topics

A stochastic golden rule and quantum Langevin equation for the low density limit

A rigorous derivation of quantum Langevin equation from microscopic dynamics in the low density limit is given. We consider a quantum model of a microscopic system (test particle) coupled with a reservoir (gas of light Bose particles) via interaction of scattering type. We formulate a mathematical procedure (the so-called stochastic golden rule) which allows us to determine the quantum Langevin equation in the limit of large time and small density of particles of the reservoir. The quantum Langevin equation describes not only dynamics of the system but also the reservoir. We show that the generator of the corresponding master equation has the Lindblad form of most general generators of completely positive semigroups

On Quantum Markov Chains on Cayley tree II: Phase transitions for the associated chain with XY-model on the Cayley tree of order three

In the present paper we study forward Quantum Markov Chains (QMC) defined on a Cayley tree. Using the tree structure of graphs, we give a construction of quantum Markov chains on a Cayley tree. By means of such constructions we prove the existence of a phase transition for the XY-model on a Cayley tree of order three in QMC scheme. By the phase transition we mean the existence of two now quasi equivalent QMC for the given family of interaction operators $\{K_{}\}$.Comment: 34 pages, 1 figur

The structure of strongly additive states and Markov triplets on the CAR algebra

We find a characterization of states satisfying equality in strong subadditivity of entropy and of Markov triplets on the CAR algebra. For even states, a more detailed structure of the density matrix is given.Comment: 11 page

Probabilità quantistica

I risultati stabiliti dallo studio assiomatico, e cioè l'esistenza di una molteplicità di modelli probabilistici empiricamente inequivalenti suggeriscono di ampliare lo scopo della teoria delle probabilità, a simiglianza di quanto è avvenuto in geometria, dallo studio di un singolo modello (quello euclideo o quello kolmogoroviano) allo studio di una molteplicit a di possibili modelli e delle loro relazioni. Mentre nella prima parte della presente esposizione ci si e limitati agli aspetti storici e, per quanto riguarda la probabilità quantistica, concettuali, il finne della presente esposizione quello di chiarire la struttura matematica della probabilità algebrica nel suo complesso e le sue relazioni con la probabilità classica. Nell'esposizione cercheremo di sottolineare come alcune delle nuove idee introdotte dalla probabilità quantistica sono emerse da motivazioni puramente matematiche ed altre dal tentativo di risolvere specifici problemi posti dalla fisica